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<html>
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<title>Holtsmark Distribution</title>
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<div class="section">
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<div class="titlepage"><div><div><h4 class="title">
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<a name="math_toolkit.dist_ref.dists.holtsmark_dist"></a><a class="link" href="holtsmark_dist.html" title="Holtsmark Distribution">Holtsmark
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Distribution</a>
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</h4></div></div></div>
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<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">holtsmark</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></pre>
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<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
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<span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a> <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
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<span class="keyword">class</span> <span class="identifier">holtsmark_distribution</span><span class="special">;</span>
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<span class="keyword">typedef</span> <span class="identifier">holtsmark_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">holtsmark</span><span class="special">;</span>
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<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
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<span class="keyword">class</span> <span class="identifier">holtsmark_distribution</span>
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<span class="special">{</span>
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<span class="keyword">public</span><span class="special">:</span>
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<span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>
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<span class="keyword">typedef</span> <span class="identifier">Policy</span> <span class="identifier">policy_type</span><span class="special">;</span>
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<span class="identifier">BOOST_MATH_GPU_ENABLED</span> <span class="identifier">holtsmark_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">location</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span>
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<span class="identifier">BOOST_MATH_GPU_ENABLED</span> <span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
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<span class="identifier">BOOST_MATH_GPU_ENABLED</span> <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
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<span class="special">};</span>
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</pre>
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<p>
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The <a href="http://en.wikipedia.org/wiki/holtsmark_distribution" target="_top">Holtsmark
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distribution</a> is named after Johan Peter Holtsmark. It is special
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case of a <a href="http://en.wikipedia.org/wiki/Stable_distribution" target="_top">stable
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distribution</a> with shape parameter α=3/2, β=0.
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</p>
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<p>
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<a href="http://en.wikipedia.org/wiki/Probability_distribution" target="_top">probability
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distribution function PDF</a> given by:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../../equations/holtsmark_ref1.svg"></span>
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</p></blockquote></div>
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<p>
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The location parameter μ is the location of the distribution, while the scale
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parameter [c] determines the width of the distribution. If the location
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is zero, and the scale 1, then the result is a standard holtsmark distribution.
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</p>
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<p>
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The distribution especially used in astrophysics for modeling gravitational
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bodies.
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</p>
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<p>
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The following graph shows how the distributions moves as the location parameter
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changes:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../../graphs/holtsmark_pdf1.svg" align="middle"></span>
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</p></blockquote></div>
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<p>
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While the following graph shows how the shape (scale) parameter alters
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the distribution:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../../graphs/holtsmark_pdf2.svg" align="middle"></span>
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</p></blockquote></div>
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<h5>
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<a name="math_toolkit.dist_ref.dists.holtsmark_dist.h0"></a>
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<span class="phrase"><a name="math_toolkit.dist_ref.dists.holtsmark_dist.member_functions"></a></span><a class="link" href="holtsmark_dist.html#math_toolkit.dist_ref.dists.holtsmark_dist.member_functions">Member
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Functions</a>
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</h5>
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<pre class="programlisting"><span class="identifier">BOOST_MATH_GPU_ENABLED</span> <span class="identifier">holtsmark_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">location</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span>
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</pre>
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<p>
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Constructs a holtsmark distribution, with location parameter <span class="emphasis"><em>location</em></span>
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and scale parameter <span class="emphasis"><em>scale</em></span>. When these parameters take
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their default values (location = 0, scale = 1) then the result is a Standard
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holtsmark Distribution.
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</p>
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<p>
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Requires scale &gt; 0, otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
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</p>
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<pre class="programlisting"><span class="identifier">BOOST_MATH_GPU_ENABLED</span> <span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
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</pre>
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<p>
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Returns the location parameter of the distribution.
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</p>
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<pre class="programlisting"><span class="identifier">BOOST_MATH_GPU_ENABLED</span> <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
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</pre>
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<p>
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Returns the scale parameter of the distribution.
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</p>
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<h5>
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<a name="math_toolkit.dist_ref.dists.holtsmark_dist.h1"></a>
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<span class="phrase"><a name="math_toolkit.dist_ref.dists.holtsmark_dist.non_member_accessors"></a></span><a class="link" href="holtsmark_dist.html#math_toolkit.dist_ref.dists.holtsmark_dist.non_member_accessors">Non-member
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Accessors</a>
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</h5>
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<p>
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All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
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functions</a> that are generic to all distributions are supported:
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<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
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<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
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<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>,
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__logcdf, __logpdf, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>,
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<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>,
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<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>,
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<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>,
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<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>. For this distribution
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all non-member accessor functions are marked with <code class="computeroutput"><span class="identifier">BOOST_MATH_GPU_ENABLED</span></code>
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and can be run on both host and device.
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</p>
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<p>
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Note however that the holtsmark distribution does not have a skewness,
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kurtosis, etc. See <a class="link" href="../../pol_ref/assert_undefined.html" title="Mathematically Undefined Function Policies">mathematically
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undefined function</a> to control whether these should fail to compile
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with a BOOST_STATIC_ASSERTION_FAILURE, which is the default.
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</p>
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<p>
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Alternately, the functions <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>,
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<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a> and
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<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>
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will all return a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
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if called.
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</p>
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<p>
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The domain of the random variable is [-[max_value], +[min_value]].
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</p>
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<h5>
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<a name="math_toolkit.dist_ref.dists.holtsmark_dist.h2"></a>
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<span class="phrase"><a name="math_toolkit.dist_ref.dists.holtsmark_dist.accuracy"></a></span><a class="link" href="holtsmark_dist.html#math_toolkit.dist_ref.dists.holtsmark_dist.accuracy">Accuracy</a>
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</h5>
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<p>
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The error is within 4 epsilon.
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</p>
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<p>
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Errors in the PDF at 64-bit double precision:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../../graphs/holtsmark_pdf_accuracy_64.png"></span>
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</p>
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<p>
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Errors in the CDF-complement at 64-bit double precision:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../../graphs/holtsmark_ccdf_accuracy_64.png"></span>
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</p>
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<h5>
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<a name="math_toolkit.dist_ref.dists.holtsmark_dist.h3"></a>
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<span class="phrase"><a name="math_toolkit.dist_ref.dists.holtsmark_dist.implementation"></a></span><a class="link" href="holtsmark_dist.html#math_toolkit.dist_ref.dists.holtsmark_dist.implementation">Implementation</a>
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</h5>
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<p>
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See references.
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</p>
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<h5>
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<a name="math_toolkit.dist_ref.dists.holtsmark_dist.h4"></a>
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<span class="phrase"><a name="math_toolkit.dist_ref.dists.holtsmark_dist.references"></a></span><a class="link" href="holtsmark_dist.html#math_toolkit.dist_ref.dists.holtsmark_dist.references">References</a>
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</h5>
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<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
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<li class="listitem">
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<a href="http://en.wikipedia.org/wiki/holtsmark_distribution" target="_top">Holtsmark
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Distribution</a>
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</li>
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<li class="listitem">
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T. Yoshimura, Numerical Evaluation and High Precision Approximation
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Formula for Holtsmark Distribution, DOI: 10.36227/techrxiv.172054657.73020014/v1,
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2024.
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</li>
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</ul></div>
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</div>
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<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
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Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
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Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
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Walker and Xiaogang Zhang<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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</p>
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